Zeno and the False Dichotomy

In defense of Parmenides’ counter-intuitive claims about the nature of the world, Zeno invented a number of arguments by which he demonstrates that the alternatives to Parmenides are, reductio ad absurdum, even more untenable. Many of Zeno’s paradoxes result from the apparently impossible conjunction of an infinite plurality within a finite interval, and Zeno took this as evidence that certain commonplace notions were misconceived: plurality, motion and space were impossible.

The arguments are strangely appealing: on the one hand we feel certain that they are unsound (and want the conclusions to be false for the sake of the world as we understand it); but on the other, we fail to grasp where Zeno has led us astray. It is not obvious that the conclusions are false, and rather exciting to imagine that the world contains the sort of mysterious contradictions that Zeno appears to demonstrate. The result has been more than 2000 years of debate during which his arguments have been ‘definitively’ rejected as simple logical fallacies—only to arise again curiously unscathed.

Two of Zeno’s most famous paradoxes are against motion: “The Dichotomy” and “The Achilles”. In the first argument, Zeno denies that motion is possible by exposing as paradoxical the apparent fact that anything in locomotion must arrive at the half-way stage before reaching its goal. The second argument denies motion by demonstrating that, given even a modest head start, a slow-moving runner can never be overtaken in a race against a speedier opponent. A third paradox, “The Argument From Finite Size”, is rather involved and the first part is no longer extant, but concludes that there could be no plurality of things in the world because such things would have to be simultaneously so small as not to have size and so large as to be unlimited.

This last argument, or at least the surviving fragment (which concludes that each of the many must be infinitely large), is similar to Zeno’s arguments against motion in that it relies on the idea of adding an infinite number of progressively smaller terms to a series that approaches a finite limit. But there is an important difference as well, and the difference makes it susceptible to the sort of refutation that nevertheless fails with respect to the other arguments.

Much of the latter half of The Argument From Finite Size survives in the work of Simplicius, who cites Zeno as claiming: “if (something without size) were added to another, it would make (the latter) no larger. For having no size, it could not contribute anything by way of size when added. And thus the thing added would be nothing.” He continues: “if (the many) exist, each one must have some size and thickness and one part (of each thing) must extend beyond the other (part of the thing). And the same reasoning holds for the projecting (part). For this too will have size and some part of it will project. Now to say this once is as good as saying it forever.” In short, Zeno is arguing that if there is a plurality of things, every thing must have parts—for that which has size has parts, and that which has no size is nothing. Moreover, each part must also be something rather than nothing and thus have size (and thereby parts) of its own. Consequently, every thing will have an infinite number of non-overlapping parts.

If this were correct, it would seem to follow that the sheer number of parts (however small each individual were in size) would (in sum) entail an infinitely large object. But the reasoning is flawed. As expected, the error is hidden in the crucial notion of infinity, which (ironically) works for and against Zeno at the same time. Zeno’s scenario has contradictory requirements. It needs the series (of parts) to be of infinite length in order to provide the multiple necessary to produce the paradoxically infinite magnitude: anything less than an infinite number of parts would result in an object of unexceptional size. At the same time, however, the final term of the series needs to be reached in order to produce the value by which the infinite number of parts will be multiplied. But an infinite series never reaches a final term; there is always a subsequent one. However long the series, the crucial final (vanishingly small) term always remains infinitely far off and thus perpetually unavailable for Zeno’s purposes. Ironically however, what is a failing in The Argument From Finite Size is a crucial element in The Dichotomy.

In the latter argument, Zeno claims that one must reach the halfway point before arriving at a destination, and the quarter-way point before arriving halfway, etc, ad infinitum. Thus, just as Zeno could never obtain the crucial final term in the infinite series of parts for The Argument From Finite Size, anyone hoping to move along a path from A to B should, according to The Dichotomy, be thwarted, even before setting out, by the ever-deferred first term of the infinite series of points that must be reached prior to the goal. The solution to the previous argument does not apply because the paradox does not rest on the magnitude of the total number of terms, but, rather, simply on the elusive first term.

But there is something about The Argument From Finite Size that sheds light on the solution to both of Zeno’s arguments against motion. A fragment of text by Melissus provides a clue: “If it had thickness, it would have parts”. It appears that mere spatial extension is a sufficient condition for the existence of “parts”. The same conception of parts is applied, in the arguments against motion, to the path an individual must travel in moving from one point to the next. Zeno, and the ancient Greeks in general, were of the opinion that spatial extension already implied a sort of delimitation into parts—and perhaps it is a reflection of our own Greek philosophical heritage that we make similar assumptions, for we accept, almost without question, the premise that one must reach the halfway point before arriving at the goal.

The result of such an assumption is that Zeno, like a magician performing his act, is able to work a sleight of hand even before we know that he has begun. And then it is too late. Having fallen into his trap and facing the absurd conclusion that motion is impossible—or at least strangely paradoxical—we look everywhere for our error but at that seemingly innocent premise. Of course we must pass the mid-point before reaching our goal; how could it be otherwise? It would be otherwise if we recognized that the mid-point needed to be created before it could be crossed. But like the Greeks, we accept the claim that the division already exists.

Nevertheless, it is possible to take a page out of Zeno’s book and construct a reductio of our own—one that highlights the absurdity of the seemingly innocent premise that the midpoint must be crossed before the goal can be reached.

The first thing to observe is that it is clearly not the halfwayness of the halfway point that matters, for the quarter-way point and the 4,194,304th-way point play equally crucial roles in the paradox. Is it, then, the pointedness of the halfway point that matters? Shall a point on the line be privileged over the line itself? And what of something in between—a line segment?

If a midway point is granted, then what reason could there be to deny that a midway interval exists (that is, not merely a dimensionless point that must be crossed but, rather, a spatially extended segment of the path), which must be passed through before the goal can be reached? If anything, the line segment is even more plausible than the point, since the only difference between the segment and the path is length—a purely contingent matter. Not only could we imagine a “path” so short as to be indistinguishable from the “segment” we could imagine a “segment” so long as to be indistinguishable from the “path”.

Though the paradox would remain virtually the same (before one can pass through the first interval one would be obliged to pass through a prior, shorter interval, and before that another, et cetera, ad infinitum), the difference is crucial—for granting the possibility that Zeno’s paradox arises even when considering an interval rather than a pointforestalls such mythico-mathematical solutions as the positing of infinitesimals, which attempt to defeat Zeno even while accepting the false premise that the midway point exists.

It is possible, perhaps, that certain mathematicians would deny the validity of such a reformulation of the puzzle, but it is not clear that Zeno would object. In fact, The Achilles paradox is virtually the same problem as The Dichotomy, but expressed in terms of intervals rather than points. In The Achilles, the eponymous hero must each moment pass through an interval equal to that traversed by a lumbering tortoise; yet, however brief the moment and however slow the tortoise, the creature nevertheless manages to travel a measurable distance and thus perpetually maintain its lead.

The next thing to observe is that it is clearly not the uni-dimensionality of the path that matters, for limiting the problem to points on a line goes not only against the spirit of the paradox (which is, after all, about things moving along a path), but also goes against other equally intuitive notions of middle. If a mid-point always exists somewhere along the uni-dimensional path from A to B, then an analogous boundary should exist on a two-dimensional surface. Consider, for instance, a blank sheet of paper. Just as someone traveling from A to B must pass through the mid-point between A and B, a line drawn down a sheet of paper must pass through the boundary that divides the top-half from the bottom.

But if that boundary exists, and the one that divides the top quarter from the bottom three-quarters (and so on ad infinitum), why not posit a boundary that divides the left half from the right? And if that one, why not a boundary that passes from the top right corner to the bottom left? It is hard to see how one could accept the notion that the mid-point between the top and the bottom of the page necessarily exists but deny that every possible division between every imaginable part of the paper is likewise equally well established. Moreover, just as it follows from the acceptance of a midway point along a line that there is a uni-dimensional interval as well, it follows that two-dimensional surfaces have two-dimensional “intervals” or segments. Since there is no reason to imagine that such segments are in anyway restricted in terms of shape and size (except, of course, for the obvious limits presented by the dimensions of the surface in question) the crude free-hand circle that I may choose to draw somewhere on the page must be considered as plausible as any other possible segment, and understood as simply making visible an already delimited part of the paper.

And we may go even further. If the line that divides a two-dimensional space always exists, should there not be an analogous boundary between the parts of a three-dimensional body as well? And just as we found no reason to privilege a straight line for the boundary separating particular parts of the two-dimensional space—so that even a crude hand-drawn circle must exist as certainly as the line that divides the top of a sheet of paper from the bottom—must we not believe that even within an apparently homogenous three-dimensional space there exist an infinity of parts in every imaginable shape and size?

Thus, in the otherwise empty corridor there must exist not only a plane dividing the north-end from the south, but something far more surprising and unlikely: the spectre of Zeno himself—or at least an area of space of precisely such size and proportion (down to the last whisker) as Zeno might have occupied the moment he bit off his own tongue and spat it at the tyrant who had him tortured to death. Those who accept the premise that the midpoint must be crossed before reaching the end of the corridor must also, it seems, accept the possibility that a defiant Zeno must be passed along the way as well.

When Zeno claims that a particular point along a path must be crossed, we are persuaded to agree. The notion strikes us as self-evidently true, presumably because “middle” is such a familiar concept and “point” such a pure mathematical entity. But if points imply intervals, and such intervals are taken out of the intangible world of mathematical lines and into the familiar three-dimensional world of our experience, and we are asked to agree that segments of every possible shape and size actually exist, and not merely that they can, in some abstract way, be delimited by an act of imagination, then it is certainly not true that their existence is self-evident.

On the contrary, most are likely to conclude that such segments of space as that horrible facsimile of a doomed Zeno exist, to the extent that they exist at all, inseparably from the minds in which they are constituted. And the same must be said of that seemingly innocent midpoint. It exists when it is imagined to exist, as does the quarter-point, the eighth-point, and the 4,194,304th-point. Much as Zeno claims, one could never begin to move along a path from A to B if one first had to pass through every one of such points—simply because the task of imagining an infinite number of possible points would be an endless process—not to mention rather beside the point.